Diffusion and Reactions in Fractals and Disordered Systems Fractal structures are found everywhere in Nature, and as a consequence anomalous diffusion has far-reaching implications for a host of phenomena. This book describes diffusion and transport in disordered media such as fractals, porous rocks, and random resistor networks. Part I contains material of general interest to statistical physics: fractals, percolation theory, regular random walks and diffusion, continuous-time random walks, and Levy walks and flights. Part II covers anomalous diffusion in fractals and disordered media, while Part III serves as an introduction to the kinetics of diffusion-limited reactions. Part IV discusses the problem of diffusion-limited coalescence in one dimension. This book written in a pedagogical style is intended for upper-level undergraduates and graduate students studying physics, chemistry, and engineering. It will also be of particular interest to young researchers requiring a clear introduction to the field. was born in 1957, in Sante Fe, Argentina, and obtained his Ph.D. in Physics from Bar-Ilan University in 1985. After a 2-year post-doctoral position in the Center of Polymer Studies at the University of Boston, he gained a permanent position at Clarkson University where he is now Associate Professor of Physics. Professor ben-Avraham has spent time as a Visiting Professor at various institutions including Heidelberg University, Bar-Ilan University, and the European Centre for Molecular Biology. He has published over 80 papers and contributed invited papers to several anthologies. DANIEL BEN-AVRAHAM
SHLOMO HAVLIN was born in 1942, in Jerusalem, Israel, and obtained his Ph.D. in 1972 from Bar-Ilan University. He stayed at Bar-Ilan University, progressing through the ranks of Research Associate, Lecturer, Senior Lecturer, and Associate Professor until in 1984 he became Professor and Chairman of the Department of Physics. He is now currently Dean of the Faculty of Exact Sciences. Since 1978 Professor Havlin has spent time as a Visiting Professor at numerous institutions including the University of Edinburgh, the National Institute of Health (USA), and Boston University. He is currently on the editorial boards of three journals. He is the author of over 400 papers and editor of ten books. He has given over 40 plenary and invited talks.
Diffusion and Reactions in Fractals and Disordered Systems Daniel ben-Avraham Clarkson University
and Shlomo Havlin Bar-Ran University
CAMBRIDGE UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Daniel ben-Avraham and Shlomo Havlin 2000 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2000 First paperback edition 2004 Typeface Times ll/14pt.
System WpX2E [DBD]
A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Ben-Avraham, Daniel, 1957Diffusion and reactions in fractals and disordered systems / Daniel ben-Avraham and Shlomo Havlin. p. cm. ISBN 0 521 62278 6 (hardback) 1. Diffusion. 2. Fractals. 3. Stochastic processes. I. Havlin, Shlomo. QC185.B46 2000 530.4'75-dc21 00-023591 CIP ISBN 0 521 62278 6 hardback ISBN 0 521 61720 0 paperback
II. Title.
to Akiva and Aliza and to Hava
Contents
Preface
page xiii
Part one: Basic concepts
1
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
Fractals Deterministic fractals Properties of fractals Random fractals Self-affine fractals Exercises Open challenges Further reading
3 3 6 7 9 11 12 12
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7
Percolation The percolation transition The fractal dimension of percolation Structural properties Percolation on the Cayley tree and scaling Exercises Open challenges Further reading
13 13 18 21 25 28 30 31
3 3.1 3.2 3.3 3.4 3.5 3.6
Random walks and diffusion The simple random walk Probability densities and the method of characteristic functions The continuum limit: diffusion Einstein's relation for diffusion and conductivity Continuous-time random walks Exercises
33 33 35 37 39 41 43
vn
viii
Contents
3.7 3.8
Open challenges Further reading
44 45
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
Beyond random walks Random walks as fractal objects Anomalous continuous-time random walks Levy flights and Levy walks Long-range correlated walks One-dimensional walks and landscapes Exercises Open challenges Further reading
46 46 47 48 50 53 55 55 56
Part two: Anomalous diffusion
57
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
Diffusion in the Sierpinski gasket Anomalous diffusion The first-passage time Conductivity and the Einstein relation The density of states: fractons and the spectral dimension Probability densities Exercises Open challenges Further reading
59 59 61 63 65 67 70 71 72
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13
Diffusion in percolation clusters The analogy with diffusion in fractals Two ensembles Scaling analysis The Alexander-Orbach conjecture Fractons The chemical distance metric Diffusion probability densities Conductivity and multifractals Numerical values of dynamical critical exponents Dynamical exponents in continuum percolation Exercises Open challenges Further reading
74 74 75 77 79 82 83 87 89 92 92 94 95 96
7 7.1
Diffusion in loopless structures Loooless fractals
98 98
Contents
ix
7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9
The relation between transport and structural exponents Diffusion in lattice animals Diffusion in DLAs Diffusion in combs with infinitely long teeth Diffusion in combs with varying teeth lengths Exercises Open challenges Further reading
101 103 104 106 108 110 112 113
8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9
Disordered transition rates Types of disorder The power-law distribution of transition rates The power-law distribution of potential barriers and wells Barriers and wells in strips (n x oo) and in d > 2 Barriers and wells in fractals Random transition rates in one dimension Exercises Open challenges Further reading
114 114 117 118 119 121 122 124 125 126
9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10
Biased anomalous diffusion Delay in a tooth under bias Combs with exponential distributions of teeth lengths Combs with power-law distributions of teeth lengths Topological bias in percolation clusters Cartesian bias in percolation clusters Bias along the backbone Time-dependent bias Exercises Open challenges Further reading
127 128 129 131 132 133 135 136 138 139 140
10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8
Excluded-volume interactions Tracer diffusion Tracer diffusion in fractals Self-avoiding walks Flory's theory SAWs in fractals Exercises Open challenges Further reading
141 141 143 144 146 148 151 152 153
x
Contents
Part three: Diffusion-limited reactions
155
11 11.1 11.2 11.3 11.4 11.5 11.6 11.7
Classical models of reactions The limiting behavior of reaction processes Classical rate equations Kinetic phase transitions Reaction-diffusion equations Exercises Open challenges Further reading
157 157 159 161 163 164 166 166
12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8
Trapping Smoluchowski's model and the trapping problem Long-time survival probabilities The distance to the nearest surviving particle Mobile traps Imperfect traps Exercises Open challenges Further reading
167 167 168 171 174 174 175 176 177
13 13.1 13.2 13.3 13.4 13.5 13.6 13.7
Simple reaction models One-species reactions: scaling and effective rate equations Two-species annihilation: segregation Discrete fluctuations Other models Exercises Open challenges Further reading
179 179 182 185 187 189 189 190
14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8
Reaction-diffusion fronts The mean-field description The shape of the reaction front in the mean-field approach Studies of the front in one dimension Reaction rates in percolation A + Bstatic -> C with a localized source of A particles Exercises Open challenges Further reading
192 192 194 195 196 200 201 202 203
Contents
xi
Part four: Diffusion-limited coalescence: an exactly solvable model 205 15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8
Coalescence and the IPDF method The one-species coalescence model The IPDF method The continuum limit Exact evolution equations The general solution Exercises Open challenges Further reading
207 207 208 211 212 213 215 216 216
16 16.1 16.2 16.3 16.4 16.5 16.6
Irreversible coalescence Simple coalescence, A + A -> A Coalescence with input Rate equations Exercises Open challenges Further reading
217 217 222 223 227 228 228
17 17.1 17.2 17.3 17.4 17.5 17.6 17.7
Reversible coalescence The equilibrium steady state The approach to equilibrium: a dynamical phase transition Rate equations Finite-size effects Exercises Open challenges Further reading
229 229 231 233 234 236 237 237
18 18.1 18.2 18.3 18.4 18.5 18.6 18.7
Complete representations of coalescence Inhomogeneous initial conditions Fisher waves Multiple-point correlation functions Shielding Exercises Open challenges Further reading
238 238 240 243 245 247 247 248
19 19.1 19.2 19.3 19.4
Finite reaction rates A model for finite coalescence rates The approximation method Kinetics crossover Finite-rate coalescence with incut
249 249 250 251 254
Contents
Xll
19.5 Exercises 19.6 Open challenges 19.7 Further reading
256 257 257
Appendix A Appendix B Appendix C Appendix D
258 260 263 266
References Index
The fractal dimension The number of distinct sites visited by random walks Exact enumeration Long-range correlations
272 313
Preface
Diffusion in disordered, fractal structures is anomalous, different than that in regular space. Fractal structures are found everywhere in Nature, and as a consequence anomalous diffusion has far-reaching implications for a host of phenomena. We see its effects in flow within fractured and porous rocks, in the anomalous density of states in dilute magnetic systems, in silica aerogels and in glassy ionic conductors, anomalous relaxation in spin glasses and in macromolecules, conductivity of superionic conductors such as hollandite and of percolation clusters of Pb on thin films of Ge and Au, electron-hole recombination in amorphous semiconductors, and fusion and trapping of excitations in porous membrane films, polymeric glasses, and isotropic mixed crystals, to mention a few examples. It was Pierre Gilles de Gennes who first realized the broad importance of anomalous diffusion, and who coined the term "the ant in the labyrinth", describing the meandering of random walkers in percolation clusters. Since the pioneering work of de Gennes, the field has expanded very rapidly. The subject has been reviewed by several authors, including ourselves, and from various perspectives. This book builds upon our review on anomalous diffusion from 1987 and it covers the vast material that has accumulated since. Many questions that were unanswered then have been settled, yet, as usual, this has only brought forth a myriad of other questions. Whole new directions of research have emerged, most noticeably in the area of diffusion-limited reactions. The scope of developments is immense and cannot possibly be addressed in one volume. Neither do we have the necessary expertise. Hence, we have chosen once again to base the presentation mostly on heuristic scaling arguments. The book is written for graduate students, and as an introduction to researchers wishing to enter the field. Much emphasis has been put on its pedagogical value. The end of each chapter includes exercises, open challenges, and references for further reading. The list of open challenges is not exhaustive. It is intended to inspire beginners (many of the challenges require computer programming, for xin
xiv
Preface
which our youngsters show remarkable aptitude), and to educate the readers to identify new directions of research. Likewise, the references given are simply those that we had at hand. Many excellent works have been left out. Nothing is implied about their relative priority or importance. We merely wished to convey a general impression of the field's scope, and to provide with some starting points. In spite of our efforts, there are bound to be misprints, inaccuracies, and outright mistakes. Please alert us to their presence by sending messages to
[email protected] (D.b-A.), or to
[email protected] (S.H.). The book is divided into four parts. Although they are closely related, they can be studied independently from one another. We ourselves have used different combinations in several graduate and upper-level undergraduate courses. Part I contains material of general interest to statistical physics: fractals, percolation theory, regular random walks and diffusion, continuous time random walks, and Levy walks and flights. Part II expands on our previous review, covering anomalous diffusion in fractals and disordered media. Part III serves as an introduction to the kinetics of diffusion-limited reactions. (The classical case of reaction-limited kinetics is briefly reviewed in Chapter 11.) By and large, the approach used in Parts II and III is that of scaling. Diffusion-limited reactions are still poorly understood, so we believe that examples of exactly solvable models are particularly important. One such example is discussed in Part IV, where we attack the problem of diffusion-limited coalescence in one dimension with the method of inter-particle distribution functions. We wish to thank our colleagues L. A. N. Amaral, J. S. Andrade, M. Barthelemy, S. Buldyrev, A. Bunde, M. A. Burschka, C. R. Doering, N. V. Dokholyan, A. L. Goldberger, P. Ch. Ivanov, P. R. King, J. Klafter, R. Kopelman, E. KoscielnyBunde, P. A. Krapivsky, H. Larralde, Y. Lee, F. Leyvraz, R. Nossal, C.-K. Peng, V. Privman, S. Redner, H. E. Roman, S. Russ, M. Schwartz, S. Schwarzer, H. Sompolinsky, H. E. Stanley, H. Taitelbaum, G. M. Viswanathan, I. Webman, and G. Weiss for years of fruitful collaborations, and we gratefully acknowledge the input of our students. Special thanks are due to Jan Kantelhardt for the beautiful cover color prints, to Roi Elran for his help with the preparation of the figures, and to S. Capelin, J. Clegg, S. Holt, and T. Fishlock, of Cambridge University Press, for their patience and for help with the technical aspects of publishing. This book could not have been written without the help of our families and friends. We thank them for their continuous encouragement and support. Daniel ben-Avraham Shlomo Havlin
Part one Basic concepts
The first part of the book includes introductory concepts and background necessary for the understanding of anomalous diffusion in disordered media. Fractals might be familiar to most readers, but their importance in modeling disordered and random media, as well as certain characteristics of the trails made by diffusing particles, makes it worthwhile to spend some time reviewing the subject. In Chapter 1 we provide working definitions of fractals, fractal dimensions, self-affine fractals, and related ideas. More importantly, we describe several algorithms for determining whether a particular object is a fractal, and for finding its fractal dimension. In the early days of fractal theory much effort was spent on merely exploring the fractal properties of various natural objects and physical models, using precisely such algorithms, and they continue to be essential tools for the study of disordered phenomena. Percolation, which is reviewed in Chapter 2, is perhaps the most important model of disordered media and of naturally occurring fractals. Percolation owes its enormous appeal to its simplicity (it can be defined and analyzed using only geometrical concepts), its remarkably wide range of applications, and its being one of the most basic models of critical phase transitions. Relevant to our purpose is the fact that studies of anomalous diffusion have traditionally focused on percolation systems, and the problem still attracts considerable interest. The percolation transition, on the other hand, gives us an excellent opportunity to introduce several useful concepts, such as critical exponents, scaling, and the upper critical dimension. In Chapter 3 we present a brief introduction to random-walk theory. Discrete random walks (in regular lattices) are discussed first, then diffusion and the diffusion equation are obtained as limiting cases. In the course of the book, we shift freely between these discrete (random walk) and continuous (diffusion)
2
Part I: Basic concepts
representations. The method of generating functions is discussed in some detail, owing to its wide applicability in other realms of statistical physics. It also eases the introduction and discussion of continuous-time random walks (CTRWs). Finally, in Chapter 4 we review other popular models of transport: Levy walks, Levy flights, and long-range correlated walks. These (and some instances of the CTRW) were originally introduced as models that exhibit anomalous transport kinetics even in regular lattices, and are therefore relatively easy to analyze, but eventually they came to be studied also in fractals and disordered lattices.
1 Fractals
Fractals model disorder in Nature far more successfully than do objects of classical geometry. In the now famous words of B. Mandelbrot: "Clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth, nor does lightning travel in a straight line" (Mandelbrot, 1982). We begin with a discussion of fractals and their most basic properties: self-similarity and symmetry under dilation, or scaling, and the fractal dimension and the ways to determine it. Our goal is to develop an intuitive understanding, and to provide some basic working tools.
1.1 Deterministic fractals Deterministic fractals are idealized geometrical structures with the property that parts of the structure are similar to the whole. While this self-similarity is a general property of fractals, it is rather a vague definition, and the best way to understand what fractals really are is through examples. Some well-known deterministic fractals are shown in Figs. 1.1-1.3. The Koch curve (Fig. 1.1) is constructed from a unit segment. The middle third section is replaced by two other segments of length | , making a tent shape, as seen in Fig. 1.1a. The same procedure is repeated for each of the four resulting segments (of length | ) . This process is iterated ad infinitum. The limiting curve is of infinite length, yet it is confined to a finite region of the plane. Thus, the Koch curve is somewhat "denser" than a regular curve of dimension d = 1, but certainly "sparser" than a two-dimensional object (its area is zero!). Intuitively, then, its dimension should be between one and two. If a regular object - such as a line segment, a square, or a cube, etc. - of dimension d is magnified by a factor b, the original object would fit bd times in the magnified one. This consideration may serve as a working definition of the fractal dimension, df (see Appendix A for more rigorous definitions). In the Koch curve, magnified by a factor of three, there
Fractals = >
(a)
(b) Fig. 1.1. The Koch curve, (a) Construction of the curve. The initiator is a unit segment. The generator replaces the middle third section by two similar sections, forming the shape of a tent. Two iterations of this process are shown, (b) The Koch curve after four iterations.
fit exactly four of the original curves. Therefore, its fractal dimension is given by 3d* = 4, or df = In4/ln3 ~ 1.262. Perhaps the most popular fractal is the Sierpinski gasket (Fig. 1.2). Here one begins with an equilateral triangle that is divided into four equal subunits, and the central subunit is discarded. Again, the process is repeated recursively. The resulting fractal dimension is given by 2d{ = 3, or df = In3/ln2 ~ 1.585. The Sierpinski sponge (also known as the Menger sponge) (Fig. 1.3) is generated from a cube that is subdivided into 3 x 3 x 3 = 27 smaller cubes. The small cube at the center and its six nearest neighbors are then discarded. The same is done with each of the remaining 20 cubes, and the process is iterated indefinitely. The limiting object has zero volume, but infinite surface area. This property is consistent with the fractal dimension of the sponge; 3df = 20, or df = In 20/ln 3 ~ 2.727, between two and three. All deterministic fractal lattices are obtained in a similar way to the examples above. Construction begins from a genus, called the initiator (e.g., the unit segment in the case of the Koch curve, an equilateral triangle for the Sierpinski gasket, etc.) and proceeds with a set of operations that are repeated indefinitely in a recursive fashion. This set of operations is called the generator. The generator may be one of two kinds. In one case, the initiator is replaced by smaller replicas of itself and the fractal builds inwardly, towards ever smaller
1.1 Deterministic fractals
(a)
(c)
Fig. 1.2. The Sierpinski gasket. Two ways for generating this fractal are shown; (a) from the outside inwards, and (b) from the inside outwards. The initiator defines an upper cutoff length of the fractal, in case (a), and a lower cutoff length, in case (b). (c) The Sierpinski gasket, shown to five generations.
length scales. The resulting fractal has then an upper cutoff length - the length of the initiator - but no characteristic microscopic length scale. In the alternative approach, replicas of the initiator are assembled into a larger object and the fractal grows outwards. The lattice then has a lower cutoff length, but no characteristic large length scale. The two methods are illustrated in Fig. 1.2. Ideal fractal lattices possess no cutoff lengths (this may be achieved by combining the two kinds of
Fractals
Fig. 1.3. The Sierpinski sponge. The initiator is a unit cube. The generator divides the unit cube into 3 x 3 x 3 smaller cubes, and seven of these subunits are removed (the one at the center of the cube, and its six nearest neighbors). The Sierpinski sponge after three iterations is shown.
growth), but real-life objects, or fractals constructed in a computer, have both upper and lower cutoffs that represent the size of the fractal structure and the size of its elementary units, respectively.
1.2 Properties of fractals A most important property of fractals is their self-similarity, or their symmetry under dilation. For example, if we examine the Koch curve (Fig. 1.1), or the Koch snowflake,as it is frequently called, we notice that there is a central object in the figure which is reminiscent of a snowman. To its right and left there are two other snowmen, each being an exact reproduction of the central snowman, only smaller by a factor of \. Both of these snowmen display even smaller replicas of themselves to their right and left, etc. In fact, if we look at the Koch curve at any given magnification (suppose we zoom in on the arm of the central snowman, say) we will see the same motif recurring again and again. The same self-similarity property can be seen also in the Sierpinski gasket (Fig. 1.2). Each triangle subunit, when it is magnified properly, is similar to the whole gasket. Note the main difference between regular Euclidean space and fractal geometries: whereas regular space is symmetric under translation, in fractals this symmetry is violated. Instead, fractals possess a new symmetry, called scale invariance, i.e., invariance under dilation. This is the property of self-similarity. Self-similarity makes fractals useful in the study of phase transitions: the state of a system at the critical transition point is also characterized by a dilation symmetry, and may be thus modeled by fractals.
1.3 Random fractals
7
The fractal dimension df is clearly not sufficient as a full description of fractal objects. Another important characteristic of fractals is their ramification. A fractal infinitely ramified if any bounded subset of the fractal can be isolated by cutting a finite number of bonds, or intersections. Thus, the Sierpinski gasket and the Koch curve are finitely ramified, whereas the Sierpinski sponge is infinitely ramified. A large variety of physical processes that take place on finitely ramified fractals can be analyzed through an exact renormalization-group approach. This accounts for the enormous popularity of the Sierpinski gasket - it is one of the simplest fractals with this beautiful property, yet it is complex enough to yield interesting insights. Many other measures and exponents may be defined to capture specific properties of fractals. For example, the concept of lacunarity is related to the degree of homogeneity of a fractal and to the extent that it is translation-invariant; the fracton, or spectral dimension, ds is an exponent describing the scaling of the density of states associated with the Laplacian operator in a fractal; and the shortest path exponent, dm[n, characterizes the length of the shortest, or chemical, path connecting two points in a fractal. Some of these properties will be required in our discussion and will be expanded upon as necessary.
1.3 Random fractals Fractals need not be generated by stiff deterministic rules. Stochastic elements may be included in the generator. Consider for example the Sierpinski carpet the surface of the Sierpinski sponge (Fig. 1.4a). It is obtained from a unit-square initiator. The generator divides this square into nine smaller cells and discards the central cell. In Fig. 1.4b we show the result obtained when the discarded cell is one of the nine cells, chosen at random. Clearly, the two figures are related, but the object in Fig. 1.4b is no longer exactly self-similar. Instead, we can argue that it is self-similar in a statistical sense: the distribution of holes looks similar at all length scales. Also, on average, the "mass" of this object (the black areas in the figure) increases by a factor of eight when space is dilated by a factor of three. Thus, the random carpet has the same fractal dimension df = In 8/ln 3 as the deterministic carpet. Generally, the mass M of random fractals scales upon dilation by a factor b as
M(bL) = bdfM(L),
(1.1a)
exactly as for deterministic fractals. Note that the solution of this functional equation is
M(L) = ALdt, where A is a constant.
(Lib)
Fractals
Fig. 1.4. Deterministic versus random fractals, (a) The deterministic Sierpinski carpet is generated from a unit square, subdivided into 3 x 3, by removing the central subunit. (b) The random Sierpinski carpet is obtained in a similar fashion, but in each generation the discarded subunit is chosen randomly.
Random fractals are useful as models for natural phenomena. Indeed, the fractal of Fig. 1.4b resembles a surface of a real sponge more closely than does the original Sierpinski carpet. A similar adaptation of the Koch curve, say, may provide an appropriate description of the coastline of Norway. Thanks to the pioneering work of Mandelbrot and others, we now know that natural objects are more likely to be fractals rather than not. How does Nature generate all these wonderful fractals? This is a very deep question and the subject of much recent research. In the next chapter we will discuss percolation - one of the most important and best understood phenomena giving rise to natural fractals. The path of a particle undergoing Brownian motion is yet another ubiquitous natural fractal that will concern us through most of this book. The fractal dimensions of random fractals are usually found numerically, either from the scaling of mass with linear size (Eq. (1.1)) - the sand-box algorithm - or from the box-counting algorithm. In the sand-box method one calculates the mass within a radius L around a site belonging to the fractal. The fractal dimension is determined by averaging over many sites, and using Eq. (l.lb). In the boxcounting algorithm the space embedding the fractal is subdivided into a hypercubic grid of cells of linear size e. One then counts the number of cells (boxes) that contain parts of the fractal, N(e). The procedure is repeated for several box sizes, and df is determined from the relation
N(e)
(1.2)
1.4 Self-affinefractals
9
The scaling of mass may also be analyzed through the density-density correlation function: c(r) = -YJP(rr)p(r'+r).
(1.3)
Here p (V) = 1 if the site r' is part of the fractal and 0 otherwise, and V = Y,rf P (r') is a normalization factor. Simply, c(r) is the average mass density around r from an arbitrary point belonging to the fractal. For isotropic self-similar fractals we expect that c{r) = c(r) ~ r~a.
(1.4)
If the fractal is embedded in d-dimensional space, its mass within a linear size L is L
/ c(r)ddr ~ Ld~a, Jo and, since M{L) ~ Ldf, it follows that a = d-d{.
(1.5)
(1.6)
The measurement of df, by any of the techniques discussed so far, requires a digitalization of the available experimental data. It is often more convenient to obtain df directly from scattering experiments. The scattering intensity is proportional to the structure factor S(q), which is simply the Fourier transform of the density c(r). For isotropic random fractals S(q) = S(q) - q-*.
(1.7)
For natural fractals, the power-law dependence of mass upon distance is valid only within the cutoff length scales A_ and A,+, and Eq. (1.7) applies only for 1/X+ < q 1.4 Self-affine fractals Until now we have considered only isotropic fractals, which display the same self-similar property in all directions. More generally, an object may possess an anisotropic dilation symmetry. In such a case we speak of self-affinity. An example is the fractal shown in Fig. 1.5. The initiator is a unit square. The generator consists of subdividing the initiator into a grid of b\ x &2 rectangular cells of which only n are retained (b\b2 — n cells are discarded). In our example bx = 3, Z?2 = 2, and n = 3. This object scales differently under dilation along the horizontal (x) and vertical (y) directions: to recover the whole fractal one must magnify a subunit by a factor of three along the x -direction, and by a factor of two along the y-direction.
Fractals
10
Fig. 1.5. A self-affine fractal. The rectangular initiator is divided into b\ x hi subunits, and n of the subunits are discarded. Three iterations of this process are shown. In this particular example b\ = 3, Z?2 = 2, and n = 3. Notice that the resulting fractal is disconnected. Designing connected self-affine fractals of a similar nature is an amusing challenge.
What is the fractal dimension of a self-affine object? If we try to apply the box-counting algorithm to the fractal of Fig. 1.5, together with Eq. (1.2), we find that the fractal dimension df seems to vary as a function of the length scale. On the one hand, as the size of the boxes decreases indefinitely, if.local = 1™
lnN(e)
In(nb1/b2)
(1.8)
6^0
This is a measure of the apparent local fractal dimension that we see as we zoom in to ever smaller length scales. On the other hand, for ever growing boxes, Zf, global
= lim
lnJV(e) _ \a(nb2/bi)
e—^oo
In bo
(1.9)
This global fractal dimension characterizes the object at the largest length scales. For an isotropic, self-similar fractal the local and global dimensions are the same and the distinction is unnecessary. Another way of characterizing the fractal dimensions of self-affine objects focuses on the anisotropic scaling of the various spatial directions, rather than on the distinction between length scales. Thus, generalizing Eq. (1.1), we may write M(bl/d?Lx, bl/dfLy)
= bM(Lx,
Ly),
(1.10)
thus defining different dimensions for the x- and j-axes. In our example M(b\Lx, b^Ly) — nM(Lx, Ly), and therefore d\ = lnn/lnb\ andd/ = lnn/ln&2Random affine fractals are also common in Nature. A mountainous landscape is the usual example, but more generally they tend to appear in phenomena involving surfaces and interfaces. The new science of surface growth profits immensely from the concept. The practical methods for measuring df of random isotropic fractals are easily generalized to affine fractals.
7.5 Exercises
II II II II
II II II II
11
II II II II
II II II II
Fig. 1.6. The Cantor set in one dimension. The initiator is the unit segment, and the generator discards the middle third subsection. Successive iterations are shown beneath each other.
1.5 Exercises 1. The Cantor set is obtained from the unit segment (0, 1) by removing the middle third (^, | ) and then repeating the process recursively (Fig. 1.6). Find the fractal dimension of this set. Generalize to d > 1 dimensions. In d = 2 and 3 dimensions the Cantor set is identical to the Sierpinski carpet and the Sierpinski sponge, respectively. (Answer: df = ln(3 J — l)/ln3.) 2. Find the box dimension of the set S = {1, \, \,...} (see Appendix A). Is this set a fractal? (Answer: ^; no.) 3. Generalize the Sierpinski gasket to d > 2 dimensions and compute the fractal dimensionality. Show in this way that a fractal may have integer df. (Answer: df = \n(d + l)/ln2; integer for d = 2n-l.) 4. Find examples of fractal lattices that have the same df but are different otherwise; for instance, finitely ramified versus infinitely ramified. Thus show that the fractal dimensionality does not characterize fractals uniquely. 5. Prove Eq. (1.7). (Hint: S(q) = f ddreiqrc(r).) 6. Construct random Cantor sets on the computer, by randomly selecting the section omitted at each generation. Try the two different methods of construction (inward and outward). Measure the fractal dimensionality using the sand-box algorithm and the scaling of mass. Notice the influence of the lower and upper cutoffs. 7. Generalize the box-counting algorithm for measuring d\ and d\ of affine fractals. Generalize the method of density-density correlations for measuring dfjocab df,giobai, d\, and d\.
8. Find a general relation between the two sets of exponents d\, d\ and df,iOcab df,giobai. (Answer: d fj ocal = 1 + d\ - d%ld\\ df,giObai = 1 + d\ - dj/d*.) 9. Starting at (x, y), move to (x +1, y — 1) or to (x + 1 , y +1) with equal probabilities. Repetition of this rule generates a random self-affine path. Simulate it on the computer and measure the various fractal dimensions. (Answer: JfjOcai = §» = 1, ^ = 1, and d\ = 2.)
12
Fractals 1.6 Open challenges
The field of fractals has been very active since about 1980, in particular concerning diffusion-limited aggregation (Meakin, 1998), fractal surfaces (Barabasi and Stanley, 1995), localization (Schreiber and Grussbach, 1991), and turbulence (Frisch, 1995; Meneveau and Sreenivasan, 1987; 1991), and finds applications in an impressive diversity of sciences: astrophysics (Labini et al., 1998a; 1998b), chemistry, geochemistry, and biophysics (Birdi, 1993), physiology (West, 1990; West and Deering, 1994; Bassinthwaighte et al., 1994), geology and geophysics (Turcotte, 1992), ecology (Sugihara and May, 1990), etc. Here are some of the many open problems. 1. What is the origin of fractals in Nature? See, for example, Bak (1996). 2. A question regarding the mere existence of fractals has been raised by Malcai etal (1997). 3. Are there complex fractal dimensions? This question has been discussed by Sornette ef a/. (1996a). 1.7 Further reading • About fractals in general and selected applications, see Mandelbrot (1977; 1982), Peitgen and Richter (1986), Peitgen et al (1992), Feder (1988), Barnsley (1988), Takayasu (1990), Stanley and Ostrowsky (1990), Vicsek (1991), Gouyet (1992), Avnir (1992), Turcotte (1992), Bak (1996), Bunde and Havlin (1994; 1996), and Meakin (1998). • Properties of fractals: ramification (Gefen et al., 1981), lacunarity (Gefen et al., 1980), the fracton dimension (Alexander and Orbach, 1982; Rammal and Toulouse, 1983), and the chemical dimension (Havlin and Nossal, 1984). • On the effect of the cutoff lengths on the scaling of fractals, with applications to galaxies, see Amici and Montuori (1998).
2 Percolation
Random fractals in Nature arise for a variety of reasons (dynamic chaotic processes, self-organized criticality, etc.) that are the focus of much current research. Percolation is one such chief mechanism. The importance of percolation lies in the fact that it models critical phase transitions of rich physical content, yet it may be formulated and understood in terms of very simple geometrical concepts. It is also an extremely versatile model, with applications to such diverse problems as supercooled water, galactic structures, fragmentation, porous materials, and earthquakes.
2.1 The percolation transition Consider a square lattice on which each bond is present with probability p, or absent with probability 1 -p. When p is small there is a dilute population of bonds, and clusters of small numbers of connected bonds predominate. As p increases, the size of the clusters also increases. Eventually, for p large enough there emerges a cluster that spans the lattice from edge to edge (Fig. 2.1). If the lattice is infinite, the inception of the spanning cluster occurs sharply upon crossing a critical threshold of the bond concentration, p = pc. The probability that a given bond belongs to the incipient infinite cluster, P^, undergoes a phase transition: it is zero for p < pc, and increases continuously as p is made larger than the critical threshold pc (Fig. 2.2). Above and close to the transition point, P^ follows a power law: Poo ~ (P - Pc)* •
(2.1)
This phenomenon is known as the percolation transition. The name comes from the possible interpretation of bonds as channels open to the flow of a fluid in a porous medium (absent bonds represent blocked channels). At the transition point the fluid can percolate through the medium for the first time. The flow rate undergoes 13
14
Percolation
L1 . i L
"U
r i
i
• ,' i I
J i
i
_
I J I I •J
i r
(b) p = 0.2
(a) p = 0.05
;
:
P
L.
j.., r
J
|
(c) p = 0.5 Fig. 2.1. Bond percolation on the square lattice. Shown are 40 x 40 square lattices, where bonds are present with probabilities p = 0.05 (a), 0.20 (b), and 0.50 (c). Notice how the clusters of connected bonds (i.e., the percolation clusters) grow in size as p increases. In (c) the concentration is equal to the critical concentration for bond percolation on the square lattice, pc = 0.5. A cluster spanning the lattice (from top to bottom) appears for the first time. The bonds of this incipient infinite cluster are highlighted in bold.
a phase transition similar to that of P^. In fact, the transition is similar to all other continuous (second-order) phase transitions in physical systems. P^ plays the role of an order parameter, analogous to magnetization in a ferromagnet, and f5 is the critical exponent of the order parameter.
2.1 The percolation transition
15
Fig. 2.2. A schematic representation of the percolation transition. The probability POQ that a bond belongs to the spanning cluster undergoes a sharp transition (in the thermodynamic limit of infinitely large systems): below a critical probability threshold pc there is no spanning cluster, so P^ = 0, but PQQ becomes finite when p > pc.
There exists a large variety of percolation models. For example, the model above can be defined on a triangular lattice, or any other lattice besides the square lattice. In site percolation the percolating elements are lattice sites, rather than bonds. In that case we think of nearest-neighbor sites as belonging to the same cluster (Fig. 2.3). Other connectivity rules may be employed: in bootstrap percolation a subset of the cluster is connected if it is attached by at least two sites, or bonds. Continuum percolation is defined without resorting to a lattice - consider for example a set of circles randomly placed on a plane, where contact is made through their partial overlap (Fig. 2.4). Finally, one may consider percolation in different space dimensions. The percolation threshold pc is affected by these various choices (Table 2.1), but critical exponents, such as /?, depend only upon the space dimension. This insensitivity to all other details is termed universality. Clearly, critical exponents capture something very essential of the nature of the model at hand. They are used to classify critical phase transitions into universality classes. Let us define some more of these important critical exponents. The typical length of finite clusters is characterized by the correlation length £. It diverges as p approaches pc as \P ~ Pc
(2.2)
Percolation
16
(a)
(b)
Fig. 2.3. Site percolation on the square lattice. Shown are 20 x 20 square lattices with sites occupied (gray squares) with probabilities p = 0.2 (a) and 0.6 (b). Nearest-neighbor sites (squares that share an edge) belong to the same cluster. The concentration in (b) is slightly above pc of the infinite system, hence a spanning cluster results. The sites of the "infinite" cluster are in black.
2.1 The percolation
transition
17
Fig. 2.4. Continuum percolation of circles on the plane. In this example the percolating elements are circles of a given diameter, which are placed randomly on the plane. Overlapping circles belong to the same cluster. As the concentration of circles increases the clusters grow in size, until a spanning percolating cluster appears (black circles). This type of percolation model requires no underlying lattice.
Table 2.1. Percolation thresholds for several two- and three-dimensional lattices and the Cayley tree. Percolation
Lattice
Bonds
Sites Triangular Square Honeycomb Face-centered cubic Body-centered cubic Simple cubic (first nearest neighbor) Simple cubic (second nearest neighbor) Simple cubic (third nearest neighbor) Cayley tree Continuum percolation d = 2 (overlapping circles) Continuum percolation d = 3 (overlapping spheres)
2sin(7r/18) a
2 bc
0.592 746 0 ' 0.697 0 4 3 J 0.198* 0.254* 0.311605^ 0.137' 0.097*" l/(z - 1) 0.312 ±0.005->' 0.2895 ± 0.0005^
ia 2
1 -2sin(7r/18)* 0.120 163 5C 0.180287 5C 0.248 812 6C'A
l/(z - 1)
-
^Exact: Essam et al (1978), Kesten (1982), Ziff (1992); *Ziff and Sapoval (1987); Lorenz and Ziff (1998); ^Suding and Ziff (1999); *Stauffer (1985a); ^Strenski et al (1991); SAcharyya and Stauffer (1998); ^Grassberger (1992a); 'Domb (1966); JVicsek and Kertesz (1981), Kertesz (1981); and ^Rintoul and Torquato (1997). c
18
Percolation
Fig. 2.5. An incipient infinite cluster. Shown is the spanning cluster in site percolation on the square lattice, as obtained from a computer simulation in a 400 x 400 square, with p = 0.6 (just above the percolation threshold). For clarity, occupied sites that do not belong to the spanning cluster have been removed, thus highlighting the presence of holes on all length scales - a characteristic feature of random fractals.
with the same critical exponent v below and above the transition. The average mass (the number of sites in site percolation, or the number of bonds in bond percolation) of finite clusters, 5, is analogous to the magnetic susceptibility in ferromagnetic phase transitions. It diverges about pc as \P-Pc
\~Y
(2.3)
again with the same exponent y on both sides of the transition. In the following sections we shall meet some more exponents and we shall see how they are related to each other.
2.2 The fractal dimension of percolation The structure of percolation clusters can be well described by fractal concepts. Consider first the incipient infinite cluster at the critical threshold. An example is shown in Fig. 2.5. As is evident, the cluster contains holes on all length scales,
2.2 The fractal dimension
19
Fig. 2.6. A schematic representation of the infinite percolation cluster above pc. The fractal features of the infinite cluster above the percolation threshold are represented schematically by repeating Sierpinski gaskets of length £, the so-called correlation length. There is self-similarity only at distances shorter than §, whereas on larger length scales the cluster is homogeneous (like a regular triangular lattice, in this drawing). similar to the random Sierpinski carpet of Fig. 1.4b. In fact, with help of the boxcounting algorithm, or other techniques from Chapter 1, one can show that the cluster is self-similar on all length scales (larger than the lattice spacing and smaller than its overall size) and can be regarded as a fractal. Its fractal dimension df describes how the mass S within a sphere of radius r scales with r: S(r)
(2.4)
S(r) is obtained by averaging over many cluster realizations (in different percolation simulations), or, equivalently, averaging over different positions of the center of the sphere in a single infinite cluster. Let us now examine percolation clusters off criticality. Below the percolation threshold the typical size of clusters is finite, of the order of the correlation length £. Therefore, clusters below criticality can be self-similar only up to the length scale of £. The system possesses a natural upper cutoff. Above criticality, £ is a measure of the size of the finite clusters in the system. The incipient infinite cluster remains infinite in extent, but its largest holes are also typically of size f. It follows that the infinite cluster can be self-similar only up to length scale £. At distances larger than £ self-similarity is lost and the infinite cluster becomes homogeneous. In other words, for length scales shorter than £ the system is scale invariant (or self-similar) whereas for length scales larger than § the system is translationally
20
Percolation
S(r)/r 2
10000 Fig. 2.7. The structure of the infinite percolation cluster above pc. The dependence of the fractal dimension upon the length scale (Eq. (2.5)) is clearly seen in this plot of S{r)/rd (d = 2) versus r, for the infinite cluster in a 2500 x 2500 percolation system. The slope of the curve is df — d for r < § ~ 200, and zero for r > §.
invariant (or homogeneous). The situation is cartooned in Fig. 2.6, in which the infinite cluster above criticality is likened to a regular lattice of Sierpinski gaskets of size § each. The peculiar structure of the infinite cluster implies that its mass scales differently at distances shorter and larger than §:
S(r)
(2.5)
Fig. 2.7 illustrates this crossover measured in a two-dimensional percolation system above pc. We can now identify df by relating it to other critical exponents. An arbitrary site, within a given region of volume V, belongs to the infinite cluster with probability S/ V (S is the mass of the infinite cluster enclosed within V). If the linear size of the region is smaller than § the cluster is self-similar, and so
Using Eqs. (2.1) and (2.2) we can express both sides of Eq. (2.6) as powers of
2.3 Structural properties __ _ r _L _,
i—i
i
i—r
—
,
r
-r
21
L _L _,
—
II
i
r
~r
-•
i l l
Fig. 2.8. Subsets of the incipient infinite percolation cluster. The spanning cluster (from top to bottom of the lattice) in a computer simulation of bond percolation on the square lattice at criticality is shown. Subsets of the cluster are highlighted: dangling ends (broken lines), blobs (solid lines), and red bonds (bold solid lines).
P~
(P ~ Pcf - (P ~
(2.7)
hence (2.8) Thus, the fractal dimension of percolation is not a new, independent exponent, but depends on the critical exponents /3 and v. Since fi and v are universal, df is also universal!
2.3 Structural properties As with other fractals, the fractal dimension is not sufficient to fully characterize the geometrical properties of percolation clusters. Different geometrical properties are important according to the physical application of the percolation model. Suppose that one applies a voltage on two sites of a metallic percolation cluster. The backbone of the cluster consists of those bonds (or sites) which carry the electric current. The remaining parts of the cluster which carry no current are
22
Percolation
Fig. 2.9. The hull of percolation clusters. The external perimeter (the hull) is highlighted in bold lines in this computer simulation of a cluster of site percolation in the square lattice. The total perimeter includes also the edges of the internal "lakes" (not shown). the dangling ends (Fig. 2.8). They are connected to the backbone by a single bond. The red bonds are those bonds that carry the total current; severing a red bond stops the current flow. The blobs are what remains from the backbone when all the red bonds are removed (Fig. 2.8). Percolation clusters (in the self-similar regime) are finitely ramified: arbitrarily large subsets of a cluster may always be isolated by cutting a finite number of red bonds. The external perimeter of a cluster, which is also called the hull, consists of those cluster sites which are connected to infinity through an uninterrupted chain of empty sites (Fig. 2.9). In contrast, the total perimeter includes also the edges of internal holes. The hull is an important model for random fractal interfaces. The fractal dimension of the backbone, dfB, is smaller than the fractal dimension of the cluster (see Table 2.2). That is to say, most of the mass of the percolation cluster is concentrated in the dangling ends, and the fractal dimension of the dangling ends is equal to that of the infinite cluster. The fractal dimension of the backbone is known only from numerical simulations. The fractal dimensions of the red bonds and of the hull are known from exact arguments. The mean number of red bonds has been shown to vary with p as (N) ~ (p — pc)~l ~ £ 1/v , hence the fractal dimension of red bonds is dred = 1/v. The fractal dimension of the hull in d = 2 is 3, however, the mass of the hull is believed to be proportional to the mass of the cluster, and both have the same fractal dimension.
23
2.3 Structural properties V " ,t
Fig. 2.10. Chemical distance. The chemical path between two sites A and B in a twodimensional percolation cluster is shown in black. Notice that more than one chemical path may exist. The union of all the chemical paths shown is called the elastic backbone. As an additional characterization of percolation clusters we mention the chemical distance. The chemical distance, £, is the length of the shortest path (along cluster sites) between two sites of the cluster (Fig. 2.10). The chemical dimension di, also known as the graph dimension or the topological dimension, describes how the mass of the cluster within a chemical length £ scales with £:
By comparing Eqs. (2.4) and (2.9), one can infer the relation between regular Euclidean distance and chemical distance:
This relation is often written as £ ~ rdmin, where dm[n = l/vi can be regarded as the fractal dimension of the minimal path. The exponent dm{n is known mainly from numerical simulations. Obviously, dm[n > 1 (see Table 2.2). In many known deterministic fractals the chemical length exponent is either di = df (e.g., for the Sierpinski gasket) or dg = 1 (e.g., for the Koch curve). An example of an
24
Percolation
(a)
(b) Fig. 2.11. The modified Koch curve. The initiator consists of a unit segment. Shown is the curve after one generation (a), and two generations (b). Notice that the shortest path (i.e., the chemical length) between the two endpoints in (a) is five units-long.
exception to this rule is exhibited by the modified Koch curve of Fig. 2.11. The fractal dimension of this object is df = In7/ln4, while its chemical dimension is dt = In7/ln5 (or Jmin = In5/ln4). The concept of chemical length finds several interesting applications, such as in the Leath algorithm for the construction of percolation clusters (Exercise 2), or in oil recovery, in which the first-passage time from the injection well to a production well a distance r away is related to I. It is also useful in the description of propagation of epidemics and forest fires. Suppose that trees in a forest are distributed as in the percolation model. Assume further that in a forest fire at each unit time a burning tree ignites fires in the trees immediately adjacent to it (the nearest neighbors). The fire front will then advance one chemical shell (sites at equal chemical distance from a common origin) per unit time. The speed of propagation would be v
=
=
r
r
(
P P
c
y .
(2.ii)
at at Ind = 2 the exponent v(dm[n — 1) « 0.16 is rather small and so the increase of v upon crossing pc is steep: a fire that could not propagate at all below pc may propagate very fast just above pc, when the concentration of trees is only slightly bigger. In Table 2.2 we list the values of some of the percolation exponents discussed above. As mentioned earlier, they are universal and depend only on the dimension-
2.4 The Cayley tree
25
Table 2.2. Fractal dimensions of the substructures composing percolation clusters. d
2
3
4
5
6
df
91/48* 1.1307 ±0.0004* 3/4* 7/4* 1.6432 ±0.0008 / 4/3* 187/91 r
2.53 ± 0.02^ 1.374 ±0.004* 1.143 ±0.01 l " 2.548 ± 0.014* 1.87±0.03 m 0.88 ± 0.02c 2.186 ±0.002^
3.05±0.05 c 1.60 ±0.05^ 1.385 ±0.055->
3.69 ±0.02^ 1.799* 1.75±0.01->
1.9 ±0.2" 0.689 ± 0.010* 2.31±0.02 r
1.93 ±0.16" 0.571 ±0.003^ 2.355 ± 0.007r
4 2 2 4 2 1/2 5/2
drain dred
dh
dfB V X
"den Nijs (1979), Nienhuis (1982); ^Jan and Stauffer (1998). Other simulations (Lorenz and Ziff, 1998) yield r = 2.189 ± 0.002; cGrassberger (1983; 1986); J Jan et al (1985); ^Grassberger (1992a). Earlier simulations (Herrmann and Stanley, 1988) yield dmin = 1.130 ± 0.004 (d = 2); ^calculated from dmin = l/vf, ^Janssen (1985), from e-expansions; ^Coniglio (1981; 1982); 'Strenski et al (1991); ^'calculated from J red = l/v; ^Sapoval et al (1985), Saleur and Duplantier (1987); zGrassberger (1999a); m Porto et al (1997b). Series expansions (Bhatti et al, 1997) yield dfB = 1.605 ± 0.015; "Hong and Stanley (1983a); ^Ballesteros et al (1997). They also find t] = 2-y/2 = 0.0944 ± 0.0017; * Adler et al (1990); and Calculated from T = 1 + d/df. For the meaning of r, see Section 2.4. Notice also that ft and y may be obtained from the other exponents, for example: ft = v(d — df), y = /3(r — 2)/(3 — r).
ality of space, not on other details of the percolation model. Above d = 6 loops in the percolation clusters are too rare to play any significant role and they can be neglected. Consequently, the values of the critical exponents for d > 6 are exactly the same as for d = 6. The dimension d = dc = 6 is called the upper critical dimension. The exponents for d > dc may be computed exactly, as we show in the next section.
2.4 Percolation on the Cayley tree and scaling The Cayley tree is a loopless lattice, generated as follows. From a central site the root, or origin - there emanate z branches. The end of each branch is a site, so there are z sites, which constitute the first shell of the Cayley tree. From each site of the first (chemical) shell there emanate z — 1 branches, generating z(z - 1) sites, which constitute the second shell. In the same fashion, from each site of the £th shell there emanate z — 1 new branches whose endpoints are sites of the (I + l)th shell (Fig. 2.12). The £th shell contains z(z - l)l~l sites and therefore the Cayley tree may be regarded as a lattice of infinite dimension, since the number of sites grows exponentially - faster than any power law. The absence of loops in the
26
Percolation
Fig. 2.12. The Cay ley tree with z = 3. The chemical shells £ = 0 (the "origin", 0), £ = 1, and £ = 2 are shown.
Cayley tree allows one to solve the percolation model (and other physics models) exactly. We now demonstrate how to obtain the percolation exponents for d > 6. We must address the issue of distances beforehand. The Cayley tree cannot be embedded in any lattice of finite dimension, and so instead of Euclidean distance one must work with chemical distance. Because of the lack of loops there is only one path between any two sites, whose length is then by definition the chemical length £. Above the critical dimension d > dc = 6 we expect that correlations are negligible and that any path on a percolation cluster is essentially a random walk; •I, or (2.12) (cf. Eq. (2.10)). This connects Euclidean distance to chemical distance. Consider now a percolation cluster on the Cayley tree. Suppose that the origin is part of a cluster. In the first shell, there are on average (s\) = pz sites belonging to that same cluster. The average number of cluster sites in the (£ + l)th shell is to+i) = (st)p(z - 1). Thus,
= Z(z -
= zp[(z - \)p ~l
(2.13)
From this we can deduce pc: when £ —• oo the number of sites in the £th shell tends to zero if p(z — 1) < 1, whereas it diverges if p(z — 1) > 1; hence 1 (2.14) Pc = z-l For p < pc, the density of cluster sites in the £th shell is
2.4 The Cayley tree
27
Therefore the correlation length in chemical distance is (using Eqs. (2.13) and (2.14))
The correlation length in regular space is £ ~ ^ , and therefore ?~(Pc-p)"1/2,
(2.16)
or v = i. The mean mass of the finite clusters (below pc) is oo
5
i .
= 1 + E to) = ^
= ^ - />ry,
(2.17)
i^i Pc- P which yields y = 1 for percolation on the Cayley tree. Consider next sns, the probability that a given site belongs to a cluster of s sites. The quantity ns is the analogous probability per cluster site, or the probability distribution of cluster sizes in a percolation system. Suppose that a cluster of s sites possesses t perimeter sites (empty sites adjacent to the cluster). The probability of such a configuration is ps(l — p)1. Hence, s,tPs(l-p)\
(2.18)
where gStt is the number of possible configurations of ^--clusters with t perimeter sites. In the Cayley tree all s-site clusters have exactly 2 + (z — 2)s perimeter sites, and Eq. (2.18) reduces to ns(p) = gspsa-p)2+{z-2)s,
(2.19)
where now gs is simply the number of possible configurations of an ^-cluster. We are interested in the behavior of ns near the percolation transition. Expanding Eq. (2.19) around pc = l/(z — 1) to lowest order in p — pc yields ns(p) - ns(pc) exp[-(p - pc)2s].
(2.20)
To estimate ns(pc) we need to compute gs, which can be done through exact combinatorics arguments. The end result is that ns behaves as a power law, Ks(Pc) ~ S~T, with T = | . The above behavior of ns is also typical of percolation in d < 6 dimensions. Generally, ns~s-Tf((p-pc)sa),
(2.21)
where f(x) is a scaling function that decays rapidly for large |JC|. Thus ns decays as s~T until some cutoff size s* ~ \p — pc\l^a, whereupon it quickly drops to zero. For percolation in the Cayley tree f(x) is the exponential in Eq. (2.20), and so
28
Percolation
We will now use the scaling form of ns to compute r in yet another way. To this end we re-compute the mean mass of finite clusters, S, in terms of ns. Since sns is the probability that an arbitrary site belongs to an ^-cluster, J2 sns — P (P < Pc)The mean mass of finite clusters is
S = % ^ ^ ~ - £>V
~ (pc - p)-W,
(2.22)
where we have used the scaling of ns (and of the cutoff at s*), and we assume that T < 3. By comparing this to Eq. (2.17) one obtains the scaling relation Y= — .
(2.23)
a
For percolation in the Cayley tree we see that r = | (consistent with the assumption that r < 3). Finally, let us compute the order-parameter exponent /3. Any site in the percolation system is (a) empty, with probability 1 — /?, (b) occupied and on the infinite cluster, with probability pPoo, or (c) occupied but not on the infinite cluster, with probability p(l — P^). Therefore,
For p < pc all clusters are finite, Yl sns — P> and Poo = 0. Above criticality J2 sns is smaller than /?, because there are occupied sites that belong to the infinite cluster. The correction comes from the upper cutoff of the sum at s = s*; J2 sns ~ J2°* sl~T ~ p- [constant x (p - pc)~(2~r)/(Jl We then find the scaling relation (2.25) o
and so /? = 1 for percolation on the Cayley tree and for percolation in d > 6. In closing this chapter, we would like to mention that there exist several variations of the percolation model that lie in different universality classes than regular percolation. These include directed percolation, invasion percolation, and long-range correlated percolation.
2.5 Exercises 1. Simulate percolation on the computer, following the simple-minded method of Section 2.1. Devise an algorithm to find out whether there is a spanning percolation cluster between any two sites, and to identify all the sites of the incipient percolation cluster.
2.5 Exercises
29
2. The heath algorithm. Percolation clusters can be built one chemical shell at a time, by using the Leath algorithm. Starting with an origin site (which represents the chemical shell 1 = 0) its nearest neighbors are assigned to the first chemical shell with probability p. The sites which were not chosen are simply marked as having been "inspected". Generally, given the first t shells of a cluster, the (i + l)th shell is constructed as follows: identify the set of nearest neighbors to the sites of shell L From this set discard any sites that belong to the cluster, or which are already marked as "inspected". The remaining sites belong to shell (1 + I) with probability p. Remember to mark the newly inspected sites which were left out. Simulate percolation clusters at p slightly larger than pc and confirm the crossover of Eq. (2.5). 3. Imagine an anisotropic percolation system in d = 2 with long range correlations, such that the correlation length depends on direction:
Generalize the formula df = d — /3/v for this case. (Answer: d\ = 1 + (vy — p)/vx;dj = l + (vx -P)/vy.) 4. From our presentation of the Cayley tree it would seem that the root of the tree is a special point. Show, to the contrary, that in an infinite Cayley tree all sites are equivalent! 5. Show that, in the Cayley tree, an ^-cluster has exactly 2 + (z — 2)s perimeter sites. (Hint: prove it by induction.) 6. The exponent a is defined by the relation Ylsns ~ \P ~ Pc\2~a- I n thermodynamic phase transitions, a characterizes the divergence of the specific heat. Show that 2 - a = (r - \)/a. 1. The critical exponent 8 characterizes the response to an external ordering field h. For percolation, it may be defined as ^ s nse~hs ~ h1^8. Show that 8 = l/(T-2). 8. The exponents a, ft, y, and 8 can all be written in terms of a and r. Therefore, any two exponents suffice to express the others. As an example, express a, 8, a, and r as functions of ft and y. 9. Percolation in one dimension may be analyzed exactly. Notice that only the subcritical phase exists, since pc = 1. Analyze this problem directly and compare it with the limit of percolation in the Cayley tree when z -> 2. 10. Define the largest cluster in a percolation system as having rank p = 1, the second largest p = 2, and so on. Show that, at criticality, the mass of the clusters scales with rank as s ~ p~dldf.
30
Percolation
2.6 Open challenges Percolation is the subject of much ongoing research. There remain many difficult theoretical open questions, such as finding exact percolation thresholds, and the exact values of various critical exponents. Until these problems are resolved, there is a point in improving the accepted numerical values of such parameters through simulations and other numerical techniques. Often this can be achieved using wellworn approaches, simply because computers get better with time! Here is a sample of interesting open problems. 1. The critical exponents ft and v are known exactly for d = 2, due to the relation of percolation to the one-state Potts model. However, no exact values exist for p and v in 2 < d < 6, nor for dfB and dm[n in all 1 < d < 6. Also, is x/t1/2.
3.4 Einstein's relation 1/2
1/2
1/2
39 1/2
1/2 Fig. 3.3. Hopping rates of a simple RW in one dimension: Eq. (3.16) is written from inspection of this diagram. where V 2 is the ^-dimensional Laplacian operator.
3.4 Einstein's relation for diffusion and conductivity Consider an asymmetric walk, in which the probabilities of stepping to the right and left are ^ + e and ^ — e, respectively. This results in an overall drift of the walk to the right. Following the same procedure as above, one obtains the diffusion equation |-P(x, 0 = -v ^-P(x, t) + D -^P(x, dt d d2
t),
(3.18)
2 / as before, and v = {Ax}/At = lea/r is the drift velocity of where D = a 2 /(2r), the walker. The diffusive motion is then simply superposed on the constant drift. Hence Eq. (3.18) can also be written as a continuity equation;
-P(x,t) = - — J(x,t), (3.19) dt dx where J = vP — D dP /dx has the meaning of probability current. The continuity equation expresses the conservation of probability, resulting from the fact that walkers cannot be created or destroyed. In Fig. 3.4 we show numerical data for (r2 ) of diffusion with a drift. Notice that the problem may be solved exactly (cf. Exercise 4), but it provides an opportunity for presenting a further example of a scaling analysis. Let the walkers now be electric charges e, in a chunk of metal (Fig. 3.5). The charges undergo diffusion, characterized by some diffusion constant D. In the presence of an electric field E, the charges attain a constant terminal velocity v given by Ohm's law: nev — —oE, where n is the density of charges per unit volume and a is the dc conductivity of the medium. If the metal is restricted to the half-space x > 0, and the field E is in the positive .*-direction, the charges will
Random walks and diffusion
40 10"
8=0.01 8=0.008 8=0.006 8=0.002 8=0
10°
/
(a)
10°
10"
10u
10 4
10'
10 5
• 8 = 0.01 A 8=0.008 0 8=0.006 O 8=0.002
/t
A
/
o
10'
A
(b)
• •
iou is-
10"'
10u
10"
/
/
10'
2
8t Fig. 3.4. A simple RW with drift, (a) The mean-square displacement for various values of the asymmetry parameter €. The asymptotic slope is denoted by the broken line of slope 1. (b) Scaling analysis of the displacement. The analytical solution of the problem yields (r2) = €2t2.
5.5 Continuous-time random walks
41
E
, v
(-)e
X
Fig. 3.5. Charges in an electric field. The electric field E imparts to the electrons a drift velocity in the negative x-direction. The current is zero at the metal's surface (at x = 0), since the charges are confined to the metal.
attain a stationary state. This may be obtained from Eq. (3.18), with dP/dt = 0:
oEdP
d2 P
x > 0,
(3.20)
and the boundary condition J(x = 0) = 0 - since there is no flux of charges into the empty region x < 0. The solution is P(x) = constant x exp[—aEx/(neD)]. On the other hand, if the metal is at temperature T, the charges would arrive at thermal equilibrium, characterized by the Boltzmann distribution PQq(x) = constant x exp[—Eex/ik^T)]. On comparing the two results, one concludes that >.
(3.21)
This is known as the Einstein relation for conductivity and diffusion. There are similar remarkable relations between other macroscopic transport parameters and the microscopic coefficient of diffusion, D.
3.5 Continuous-time random walks Montroll and Weiss (1965) introduced the concept of continuous-time random walks (CTRWs) as a way to render time continuous, without appealing to the diffusion limit. The model achieves much more than this modest goal. Some forms of CTRW are fundamentally different than the classical diffusion model, and the theory has numerous important applications. Imagine a random walk on a lattice, starting at the origin, but such that the steps are taken at random times. Let \/r(t) be the probability density for the waiting time
42
Random walks and diffusion
between successive steps. We assume that the waiting times for different jumps are statistically independent and are all characterized by i/s(t). The probability that the waiting time between steps is greater than t is
= j"f(tf)dt'.
(3.22)
Define also ijrn(t) as the probability density that the rath jump occurs at time t. Clearly, x/r\(t) = tyit), and, since the waiting times between steps are independent,
= ff
- t')dt!.
(3.23)
Using the Laplace transform f f{t)e~st dt Jo (and likewise for other functions), we find jr(s)=
xlrn(s) = f{s)\
*!>(*)
(3.24)
1 - xjr(s) ^ . s
(3.25)
Given now Pn (r) - the probability of being at r at the n\h step - one can express
P(r,t) as OO
ntt
P(r, t) = Y] Pn(r) // fn(t')V(t - tf) dt'.
(3.26)
JJ
°
n=0
The integral represents the probability that, once the walker had arrived at site r at time tf < t, it would remain there until time t. Taking the Laplace transform, we obtain (3.27) S
n=0
Finally, we take the Fourier transform, and, using the result of Eqs. (3.6) and (3.9), the infinite sum may be carried out explicitly: , s) =
I — xlr(s) ^ - ^
.
(3.28)
One can now invert the double transform to obtain the distribution itself. The moments are somewhat easier to obtain. For example, in one dimension:
. dP(k,s) (r(s)) = -i dk
k=o
s[l-\jr(s)Y
(3.29)
3.6 Exercises
43
and v
'
-,
dk2
(3.30)
where \xn = f rnp(r) dr are the moments of the step distribution function. The special case oftyit) = 8(t — r) (r constant) reduces to the regular random walks discussed earlier and confirms the CTRW formalism. In fact, any \j/{t) that falls off fast enough yields similar regular behavior. However, interesting anomalous behavior may be obtained with slow-decaying \jr, as will be shown in the following chapter. 3.6 Exercises 1. Compute X(k) of Eq. (3.7) for a one-dimensional random walk (RW). 2. Consider a RW in a ^-dimensional hypercubic lattice with lattice spacing a. In a time step x the walker steps to one of the 2d nearest sites with probability e/(2d), or stays put with probability 1 — 6. Write a master equation (similar to (3.16)) and obtain the continuum limit, Eq. (3.17). What is the diffusion coefficient D(e)l (Answer: D = ea2/(2dx).) 3. Verify that P(r, t) of Eq. (3.15) solves the diffusion equation (3.17), by direct substitution. Obtain the solution explicitly, for when the initial condition is P(JC,0) =
8{x).
4. Asymmetric RWs. Consider an asymmetric random walk in one dimension, with / rp(r)dr = b > 0 and f(r — b)2p(r)dr = a2. Apply the method of characteristic functions and show that P(r, t) of Eq. (3.15) generalizes to
P(r,t) = (4jrD0~1/2exp[(r - vt)2/(4Dt)l where D = a2/(2x) and v = b/x. Verify that this distribution is a solution to Eq. (3.18). 5. Persistent RWs. A random walker in one dimension has probability a of continuing in the same direction as the previous step (and probability /3 = 1 —a of reversing direction). Let the probability that the walker is at m at step n be Pn(m), when it is coming from m — 1; and Qn(m), when it is coming from m + 1. Generalize the characteristic-function approach of Section 3.2 by considering the state vector (Pn(m), Qnim)). The structure function may then be written as a matrix. Work out the long-time asymptotic limit and show that it is similar to a regular random walk, without persistence. 6. Derive recursion relations for Pn(ni) and Qn(jn) of the persistent RW. Pass to the continuum limit in the usual way, writing a = I — r/(2T), where T is a constant. For U = P + Q, obtain the telegrapher's equation
d2u dt2
i
\du T dt
L r
2
d2u dx2
44
Random walks and diffusion
7. Motion in a fluid medium is hindered by a drag force proportional to the speed of motion: F drag = —yMv. M is the mass of the moving object, and y is the macroscopic drag coefficient. Derive the Einstein relation between y and the microscopic diffusion coefficient D. (Hint: consider motion under an external constant force, such as gravity.) (Answer: y~l = MD/(kBT).) 8. The regular RW is a special case of the CTRW, with a waiting-time distribution function yf/(t) = 8(t — r). Verify this statement by computing (r), (r 2 ), and P(r, t) through the CTRW formalism. 9. Given a CTRW with f{t) = (l/T)exp(-t/T), where T is a constant, compute (r) and (r2) as functions of time. Show that the probability density
is P(r,t) = exp(-t/T)Ir(t/T)d,
where Ir(z) = [1/(2TT)] / ^ exp(-i>0 -
z cos 0)d6 is a Bessel function of imaginary argument. Obtain the limit t -> oo of P(0, t) ~ VT/(2nt). Compare this result with the simple random walk. 10. Generalize the approach of Section 3.5 to higher dimensions, and obtain formulae analogous to (3.29) and (3.30).
3.7 Open challenges 1. The number of distinct sites visited by a walker in a lattice up to time t, S(t), is an important quantity for many applications (Appendix B). The full asymptotic distribution of S(t) is known to be Gaussian in three dimensions, with a mean t and variance t log t\ and also in higher dimensions it is Gaussian. The scaling of the first moment is known in all dimensions: {S(t)) ~ t1^2 (d = 1), //log t (d = 2), and t (d > 3). The second moment of S(t) has also been derived (Larralde and Weiss, 1995). However, the full distribution of S(t) remains unknown. 2. The problem of self-attracting (or self-repelling) random walks has not yet been resolved. In this case the RW has a higher (or lower) probability of visiting previously visited sites. Open problems include the derivation of the mean-square displacement and the number of distinct sites. The scaling of these quantities with time is controversial (Lee, 1998; Sapozhnikov, 1998). The limit in which the probability of revisiting previously visited sites is zero is the well-known problem of self-avoiding walks (SAWs). 3. The paths generated by random walks on lattices are intricate geometrical objects of which little is known. For example, the scaling of the chemical distance with the Euclidean distance has not been studied. A partial study of random-walk paths in two dimensions was performed by Movshovitz and Havlin (1988). The question of the distribution of loops in random-walk trails has recently been introduced by Wolfling and Kantor (1999).
3.8 Further reading
45
4. The problem of N diffusing walkers when a fraction p of the walkers can die and a fraction q gives birth has been studied by Meyer et al (1996b). The mean displacement of the walkers with respect to the center of mass has been found forp = q to reach a plateau as a function of time, i.e., the walkers tend to cluster. The number of distinct sites visited by this group (the territory) has not been studied. Moreover, the known solution (Meyer et al, 1996b) is valid only when p and q are fixed fractions, rather than probabilities. 3.8 Further reading The theory of random walks: Chandrasekhar (1943), Spitzer (1976), Barber and Ninham (1970), Weiss and Rubin (1983), Montroll and Shlesinger (1984), Weiss (1994), and Hughes (1995). Random walks in biology: Berg (1993). Brownian motion: Lavenda (1985). The number of distinct sites visited by N diffusing particles in d dimensions has been studied analytically by Larralde et al (1992a; 1992b). Selected applications of random walks: optical imaging (Bonner et al, 1987) and amorphous disordered systems (Bunde and Havlin, 1996). Diffusion in condensed matter: Karger et al (1998). Random walks contributed to the field of crystallography (Klug, 1958). The development of this theory has been recognized in the awarding of the Nobel Prize to Hauptman and Karle in 1985. An introduction to transport models based on CTRWs: Scher and Lax (1973) and Scher and Montroll (1975).
4 Beyond random walks
Random walks normally obey Gaussian statistics, and their average square displacement increases linearly with time; (r2) ~ t. In many physical systems, however, it is found that diffusion follows an anomalous pattern: the mean-square displacement is (r2) ~ t2^dw9 where dw ^ 2. Here we discuss several models of anomalous diffusion, including CTRWs (with algebraically long waiting times), Levy flights and Levy walks, and a variation of Mandelbrot's fractional-Brownianmotion (FBM) model. These models serve as useful, tractable approximations to the more difficult problem of anomalous diffusion in disordered media, which is discussed in subsequent chapters.
4.1 Random walks as fractal objects The trail left by a random walker is a complicated random object. Remarkably, under close scrutiny it is found that the trail is self-similar and can be thought of as a fractal (Exercise 1). The ubiquity of diffusion in Nature makes it one of the most fundamental mechanisms giving rise to random fractals. The fractal dimension of a random walk is called the walk dimension and is denoted by dw. If we think of the sites visited by a walker as "mass", then the mass of the walk is proportional to time. We can then write M~t~rd™,
(4.1)
where r is the typical distance covered after time t. The mean-square displacement is then given by (r 2 (0> ~t2/d™.
(4.2)
For regular diffusion dw = 2, but in fractals dw ^ 2 and one then talks of anomalous diffusion. Next, we shall use the CTRW model to illustrate a different 46
4.2 Anomalous CTRWs
47
cause for anomalous diffusion: slow-decaying waiting times between consecutive steps.
4.2 Anomalous continuous-time random walks We have seen that, when the waiting-time density function of a CTRW has a characteristic time scale (i.e., a finite first moment), the mean-square displacement is proportional to t and the probability density is Gaussian. Here we consider the case in which the waiting-time distribution possesses no characteristic time scale and show that this leads to anomalous diffusion. Consider a CTRW with a power-law waiting-time distribution f(t) - A r ( y + 1 ) ,
0 < y < 1.
(4.3)
(The exponent y, used here, and p, in Section 4.3, should not be confused with the Y a n d P of percolation theory and critical phase transitions.) Such a situation occurs for example when at each lattice site the walker is subject to a potential V, distributed exponentially, P(V) = Voexp(—V/Vo). In a three-dimensional lattice a random walker hardly ever revisits a previously visited site and therefore the waiting times between steps are practically independent, just like in a CTRW. The probability that the walker exits a well of depth V is proportional to the Boltzmann factor w ~ exp(—PV), where f5 = \/{k^T). The distribution of transition rates would then be (p(w) = P(V) dV/dw ~ W(l/Vo-P)/P9 anc j hence the waiting time - which is proportional to l/w - is distributed as in Eq. (4.3), with Y = k^T/ Vb(The restriction on y can always be met by lowering the temperature.) Such a system may model the thermal relaxation of complex molecules such as large proteins, or it may mimic dynamic processes in glasses. The mean-square displacement can be found in an analogous way to Eq. (3.30): \
(4.4)
By comparing this to (4.2) we conclude that diffusion is anomalous, with dw = 2/y. It is also instructive to derive the probability density of the walk, P(r,t),in the long-time asymptotic limit. One finds
This generalizes the result for regular diffusion, where 8 = 2 (Eq. (3.15)). As with other fractal objects, the fractal dimension dw does not fully characterize random walks. For example, it is not completely clear whether 8 depends solely on dw, or whether there are cases in which 8 is a new independent exponent. Many other properties of random walks are studied with different applications in mind.
48
Beyond random walks
Examples are the span of the walk, which is the distance between the leftmost and rightmost points visited by a walker;first-passagetimes, i.e., the time until a certain set of points is visited for the first time; and survival probabilities - the chances that a walker has avoided a given subset of sites up to a certain time. A useful quantity in many applications is the number of distinct sites visited by a random walker (Appendix B).
4.3 Levy flights and Levy walks In many natural realizations it was found that diffusion is enhanced and (r2) scales faster than linearly with time. One of the important models for enhanced diffusion is Levy flights and its generalization, called Levy walks (Shlesinger et al, 1987). Consider a random walker that at each time step t jumps in some random direction (taken from a uniform distribution) to a distance r, taken from a powerlaw distribution; p(r) ~ l/r 1 + ^.
(4.6)
It is implied by Eq. (4.6) that the second moment of p{r) diverges for fi < d (d is the system's dimension), or that the Levy jumps have no characteristic length scale. Figure 4.1 shows some typical 1000-step Levy flights. It can be shown that, for /3 < d, the probability density of the walker being at r at time t is not Gaussian but rather follows the Levy distribution P(r,t)~t/rl+P,
(4.7)
and the mean-square displacement diverges. The form of the probability density, when it is transformed from real space r to Fourierfc-space,is P(k,t) = exp(-t\k\f>).
(4.8)
The fractal dimension of the sites visited by a Levy flight is df = $ for fi < 2 and df = 2 for p > 2. The largest jump after t trials, rmax, can be estimated by comparing the distance probability-density function with a uniformly distributed variable u: p{r) dr = du/t, or r~$ ~ u. The minimal value of u after t trials is l/t, hence rmax ~ t1^. This relation also represents the first-passage time for a Levy flight to reach a distance r; t ~ r&. In either case, the scaling implies a fractal dimension of the Levy flight of df = ft (/3 < 2). Similar analytical arguments are used in Chapter 8. Levy flights have been found useful for describing a wide range of physical phenomena, including chaotic diffusion in Josephson junctions (Geisel etal, 1985) and turbulent diffusion (Shlesinger and Klafter, 1986). Levy flights were observed recently in several biological systems: foraging ants and Drosophila flies perform
4.3 Levy flights and Levy walks
49
(a)
Fig. 4.1. Trajectories of typical Levy flights in d = 2 for several values of f$: (a) ft = 2, (b) 0 = 3, and (c) 0 = 6.
Levy flights (Cole, 1995); large birds such as the wandering albatross follow a power-law distribution of flight-time intervals (Viswanathan et al, 1996); and a model for evolution of DNA based on Levy flights has been suggested by Buldyrev
50
Beyond random walks
et ah (1993). Levy flights have been found useful even in economics (Mantegna and Stanley, 1995). When the time to make a step r is assumed to be proportional to r (constant velocity) the resulting model is called a Levy walk. One way to introduce this velocity is by using a coupled spatio-temporal probability density i/r(r,t) for the random walker performing a displacement r at time t (Klafter et al, 1996): f(r,t) = f{r\t)p{r).
(4.9)
Here \/f(r\t) is the conditional probability that a jump of size r takes a time t. For random walks moving with a velocity v,\j/(r\t) = 8(t — \r\/v). Note that in general the velocity need not be constant. Finite velocities lead to a finite mean-square displacement at any given time t. For constant velocity it can be shown that the mean-square displacement depends on j5 as t\ t2/\nt, (r2) = • t3~P, tint, t,
0 I). Show that the probability density of the rescaled step is also Gaussian. How should one rescale space, so that the RW is invariant (that is, P\rf, t') and p'(rf) are similar to P(r, t) and p(r) of the original walk)? Argue that, in the long-time asymptotic limit, the same self-similarity is observed for other forms of p(r). 2. Calculate the number of distinct sites visited by a Levy flight in a onedimensional system and in a d-dimensional system. See Gillis and Weiss (1970). 3. Use Eqs. (3.30) and (4.3) to derive the anomalous behavior cited in Eq. (4.4). 4. Derive the behavior of Levy walks cited in Eq. (4.10).
4.7 Open challenges 1. The form of P(r, t) in the CTRW model in the regime r 2 for anomalous diffusion in fractals. The number of fractal lattice sites within a radius L is Ldf. On the other hand, a walker must perform t ~ Ldw steps to attain a r.m.s. displacement L. It follows that, if dw > df, each site within the radius L would be visited several times (the number of visits per site would increase with the radius as Ldw~df). In this case the trail of the walk is "compact" - the number of distinct sites visited equals the volume Ld{. In the opposite case of dw < df, the walker cannot visit all lattice sites within L because their number exceeds the number of steps. The trail is tenuous: only a vanishing fraction Ld™~df of the sites is visited. The compact walks are also called recurrent, because of their property that, in the long-time limit, a walker returns to the origin with probability unity. For nonrecurrent walks with dw < df the probability of return to the origin vanishes with increasing time. In regular Euclidean space dw = 2 and df — d, hence diffusion in one dimension is recurrent and diffusion in d > 3 is nonrecurrent. The case of d = 2 is marginal, dw = df, and is accompanied by logarithmic divergences.
5.2 The first-passage time We now compute dw for the Sierpinski gasket, following an exact real-space renormalization-group procedure. We focus on T, the mean first-passage time taken to traverse the lattice from the vertex at the apex to one of the remaining two vertices at the bottom (Fig. 5.2a). T can be related to Tf, the equivalent first-passage time in a lattice rescaled by a factor of two (Fig. 5.2b). Let A and B be the first-passage times from the inner vertices of the rescaled lattice to the lower O vertices (Fig. 5.2b). Then, exploiting the finite ramification of the gasket, we see that T = T + A, A =T + \A + \B + \T',
(5.1)
B =T + \A. For example, suppose that a walker is in one of the nodes denoted by A in Fig. 5.2b. If it gets first to the vertex at the top, it will then take an additional time T' to exit
62
The Sierpinski gasket
Fig. 5.2. Rescaling of first-passage times: (a) The mean first-passage time for traversing a Sierpinski gasket from the top apex to either of the vertices at the base of the gasket (marked by 0) is T. (b) When the lattice is rescaled by a factor of two, the first passage time is T\ which is clearly larger than T. A and B are thefirst-passagetimes for exit through 0, for walks originating from the junctions between the rescaled subunits. A, B, and most importantly, Tr may be related to T exactly (Eq. (5.1)). through O. If instead it gets first to the A node, it will take an extra time A to exit through O. If it gets first to the B node, it will take an extra time B to exit. Finally, if it gets first to 0 , no additional time is necessary, for it would have already exited the unit. Since each of these possibilities happens with equal probability, and since it takes an average time T to get to any of the nearest nodes, we have A = \{T + Tf) + \(T + A) + \(T + B) + \T, which is the second equation in (5.1). The remaining equations are obtained in a similar fashion. The solution of Eqs. (5.1) is V = 5T (and A = AT and B = 3T). We can then compute dw from the way it relates time T to length L: T ~ Z>
(5.2)
(cf. Eq. (4.2)). Upon rescaling of space by a factor of two, L —• L1 — 2L, time rescales as T -> T1 = 5T. Since Eq. (5.2) holds also for the rescaled problem; T' ~ (Z/) Jw , we conclude that j ^ = 2.322 log 2
(5.3)
for random walks on the Sierpinski gasket. This compares very nicely with the numerical results of Monte Carlo simulations and exact enumeration.
5.3 Conductivity and the Einstein relation
63
5.3 Conductivity and the Einstein relation One of the main reasons for the interest in diffusion in fractals and disordered media is its relevance to other physical properties of the medium. Here we wish to discuss the connection between diffusion and electric conductivity. The basic relation is the Einstein relation derived in Section 3.4, a = 2 ne D/{k^T) (Eq. (3.21)). In anomalous diffusion the diffusion constant D is not really a constant, but actually changes as diffusion proceeds: D = ^-j^
~ y,
(5.4)
where L is the typical distance covered at time t. The carrier density n is proportional to the density of the substrate, and hence in a fractal n ~ M/Ld ~ Ldf~d. The conductivity of a fractal is expected to scale as a power of its linear size, a ~ L~A,
(5.5)
where jl is the conductivity exponent. On inserting these expressions into the Einstein relation we find
or, since t ~ Z/ w , d^ = 2-d
+ df + /l = df + t;.
(5.6)
In the last expression f is the resistance exponent which denotes the scaling of resistance with length: R~lJ.
(5.7)
The relation between jl and f is obtained from the relation between conductivity and resistance. Conductivity is defined through the equation j = GE, where j is the current density and E is the electric field. Resistance is given by Ohm's law, / = V/R, where / is the total current and V is the potential gap, V ~ EL. In d dimensions the current density is j ~ I/Ld~l (current per unit "area"). Comparison of the equations for a and R yields I = 2 — d + /x, as in Eq. (5.6). Notice that, for regular space, conductivity is an intensive quantity and jl — 0, but generally for fractals jl > 0. We now exploit the relation between resistance and random walks (Eq. (5.6)) to compute dw indirectly. Suppose that a current / is injected into the top vertex of a Sierpinski gasket and is drawn out from the other two vertices (Fig. 5.3a). We wish to compute the resistance R between the top vertex and a vertex at the bottom. Let
64
The Sierpinski gasket
R
1/2
//2
R
(a)
(b)
Fig. 5.3. Rescaling of the resistance of the Sierpinski gasket. A current / is injected at the apex of the gasket and collected from the lower vertices, (a) The effective resistance between the apex and a vertex at the bottom is R. (b) In a gasket rescaled by a factor of two the current will flow from vertex to vertex along the path indicated by the arrows. This suggests the rescaling of resistance of Eq. (5.8).
Rf be the analogous resistance in a lattice rescaled by a factor of two. Owing to symmetry the current flows as indicated in Fig. 5.3b. From this we conclude that Rf = R + (R || 2R) = §#,
(5.8)
where the vertical bars denote combination of resistances in parallel. From the scaling of resistance with length (Eq. (5.7)) we find
(5 9)
-
Using Eq. (5.6) and the fact that the fractal dimension of the Sierpinski gasket is df = Iog3/log2, we recover dw = Iog5/log2. This method can be applied to calculate I and dw for finitely ramified fractals. The remarkable relation between resistance and random walks provides us with a most useful technique for estimating £ in fractals. One simply simulates diffusion or performs exact enumeration in the fractal in question and measures dw. The fractal dimensionality which is also necessary for the computation of f (Eq. (5.7)) is related to the probability of the walker returning to the origin (see next section) and can too be measured from the same diffusion simulations. This method has been used to obtain reliable estimates of I and /x for percolation clusters, lattice animals, diffusion-limited aggregates, and other random substrates.
5.4 The density of states
65
5.4 The density of states: fractons and the spectral dimension The Laplacian operator V2 of the diffusion equation dt
appears in many other physics problems. For example, in the Hamiltonian H = — [h2/(2m)]V2 of the Schrodinger equation for a free particle:
as well as in heat equations, in Poisson's equation for electric potential, etc. As a specific example, consider the vibrational modes of an elastic fractal network consisting of particles connected by harmonic springs. In fractals the vibrational modes are called fractons rather than phonons. In the isotropic case in which the spring constants are assumed to be scalars one obtains the equations of motion
^P
J^
(5.10)
where [/,- is the displacement of the /th site, and the sum runs over all nearest neighbors j of site i. (For simplicity, we assume that all particles have unit mass.) Diffusion in a discrete fractal lattice would be described by a similar equation, except that £/,- (t) is then replaced by Pt (t) - the probability of being at site / at time t - and there is a first-order time derivative instead of the second-order derivative on the LHS of Eq. (5.10). Also, in the case of diffusion k(j represents the jump frequency for jumps from site / to site j . Notice that, when all the ktj are equal, the RHS of (5.10) is in fact a discrete form of V2C/. Equation (5.10) may be solved by a standard classical-mechanics approach. On making the substitution Ui(t) = Ui(w)exp(—ia)t) one obtains N linear homogeneous equations for the N unknowns Ui(w) (N is the total number of sites). When ktj = kji, i.e., for unbiased diffusion, the equations yield positive The general solution of (5.10) real eigenvalues: col > 0 (n = 1,2, ...,N). is Ut(t) = St{Y^=1CnV? exp(-ia)nt)}9 where (*?,..., *£) is an orthogonal set of eigenvectors and the Cn are complex constants that are determined from initial conditions. For example, if at t — 0 only the jo th particle is displaced, £7/(0) = ^ ^ , we get Ui+k{t) = For the corresponding diffusion problem, when the walks originate at / = jo, the
66
The Sierpinski gasket
solution is
W O= where en = co^ is the energy associated with the vibration mode n. The average probability of finding the walker at distance r from the origin at time t is then
where i
N
N(r)
-,
Notice the double average: over all N(r) points a distance r away from jo, and over the starting point jo. In particular, for r = 0, 1 ^
)
since the eigenvectors are normalized, O(0, n) = \/N. Passing to the continuum limit Af —>• oo, one may rewrite the last equation as (5.11) - / •
where p(e) is the energy density of states. The probability of returning to the origin at time t may also be estimated from the following argument. At time t the r.m.s. displacement is L ~ t1^™. The probability that a walker is in any of the Ldf sites within that radius is approximately uniform, so P(0, t) ~ 1/Ld{ ~ l/td{/dw. Using this relation in Eq. (5.11), wefinallyfind p(e) - e*/^" 1 = e* 72" 1 .
(5.12a)
The vibrational density of states g(co) then follows from p(e)de = g(co) dco and co'1'-1.
(5.12b)
In regular Euclidean space the two densities are p(e) ~ ed/2~l a n d g(&>) ~ ty^"1. In fractals, the vibrational modes are called fractons instead of phonons. We see that, in the case of fractals, the effective dimension which controls the density of states is ds = 2df/dw
(5.13)
rather than d. The exponent ds is known as the fracton dimension, or also as the
5.5 Probability densities
67
spectral dimension. (We use the subscript s to avoid confusion with the fractal dimension df.) The relation between diffusion and the density of states can also be derived from the following scaling argument. The density of states is an extensive quantity (proportional to the volume of the system), thus, upon rescaling of length L —> bL, the density of states rescales as pbL{t) = bdipL{6).
(5.14)
From the Schrodinger equation (we could use any phenomenon involving the V2 operator) we see that the energy - the eigenvalue of TL - is inversely proportional to time, which scales as t ~ ZA, so €bL=b-d"€L.
(5.15)
The density of states in the original system and that in the rescaled system are related through probability theory: pbhi^bh)dtbL = PL{^L)d€i, or, using Eq.(5.15), PbL(ebL) = bd-pL(eL).
(5.16)
The solution of Eqs. (5.14)-(5.16) is p(be) = bdi/dw~lp(e), in agreement with Eq. (5.12a). For fractons one can follow the same line of reasoning, except that instead of Eq. (5.15) we now have a>bL ~ b~dw^2a>L (because of the second-order time derivative, as opposed to the first-order derivative in Schrodinger's equation). The result is identical with Eq. (5.12b). Both the Schrodinger equation and the problem of fractons have been treated exactly in the Sierpinski gasket. The analyses confirm the relation to diffusion discussed above.
5.5 Probability densities In random-walk theory probability densities play a central role. Indeed, let P(r, t) be the probability density that a walker that started at the origin at time t = 0 is at point r at time t. In regular space random walks are Markovian and P(r, t) determines the diffusion process completely. Diffusion in fractals is in general nonMarkovian and, although P(r, t) does not then provide a complete characterization it remains a quantity of primary importance. For example, knowledge of it allows the computation of the moments = / rKP(r,t)dar
(5.17)
68
The Sierpinski gasket
the probability of return to the origin is simply P(0, t) and from it one may obtain the fracton dimension ds; and P(r,t) plays a prominent role in the Flory theory of self-avoiding walks. For regular diffusion P(r, t) is a Gaussian distribution (Eq. (3.15)). In the case of anomalous diffusion, one expects that P(r, t) would have a similar scaling and functional form. A reasonable assumption is rdt-d
The fundamental scaling of r ~ tdw is contained in the argument of O. The prefactor of rdf~d accounts for the expected scaling of P with fractal density. (Consider for example the case in which P is homogeneous in space.) Finally, the prefactor of t~ds/2 follows from normalization; f P ddr — 1, if ds = 2df/dw. For r 3> f 1^ we expect some form of exponential decay, which may be incorporated in the function O (Fig. 5.4): rdf-d
where 8 is a shape exponent. The smaller 8 the sharper the peak of the distribution. It has been suggested that g = 8/2 — df (Acedo and Yuste, 1998). Notice that Eqs. (5.18) and (5.19) agree with regular diffusion (where df = d, dw = 2, and 8 = 2). As a particularly simple example consider diffusion in linear polymers, modeled by self-avoiding walks (SAWs). SAWs have a random fractal structure but are topologically linear, similar to the Koch curve. Therefore, the process of diffusion along a SAW is normal, just like in one-dimensional space. Using the length £ measured along the SAW (the chemical length), the probability density 0(£, t) of diffusion in the SAW is
The relation between I and distance in regular space, r, is given through the fractal dimensionality of the SAW, rdf ~ I. However, a better description is obtained from the probability density P{r\l) that an £-step SAW section is of length r:
p(r\i) =
5.5 Probability densities
69
(a) Ot=64 Dt=128 Ot=256 At=512 Pc),
(6.5)
where we have used § ~ \p — pc\~v and fi = jlv. At criticality the diffusion coefficient is not a function of p (since p = pc is fixed) and its dependence on time follows from (r2(t)} ~ t1!^, as discussed in the previous section. Thus, D\t,pc)
~f(2-OAC.
(6.6)
In the nonconducting phase, p < pc, the dominant contribution comes from
78
Diffusion in percolation clusters
the largest clusters, of size £ ~ \p — pc\~v. At long times r 2 (0 ~ £ 2, but the probability of being in any of the large clusters is proportional to \p — pc\P. Hence, D\t -> oo, p) ~ t~\pc - p)-2v+fi,
(p < pc).
(6.7)
Equations (6.5)-(6.7) may be combined in the scaling form: D\t, p) = tV oo, D ~ t~l(pc — p)~lv• The scaling function of D that is consistent with these properties is D(t, p) =
tV^-ig^t(d
where
g(x)
cM ^ constant (—x)~
lv
as x -> oo, asx->0, as x —>• — o o .
(6.13)
6.4 The Alexander-Orbach conjecture
79
Table 6.1. Dynamical exponents for percolation.
2 3 4 5 6
dw
ds
jl
dyw
2.878 ± 0.001" 3.88 ± 0.03c 4.68 ± 0.08* 5.50 ± 0.06*
1.318 ±0.001" 1.32±0.01c 1.30 ±0.04^ 1.34 ±0.02'
0.9826 ± 0.0008" 2.26 ± 0.04^ 3.63 ± 0.03^ 4.81 ± 0.04^
2.62 ± 0.03^ 3.09 ± 0.03^
6
4 3
6
4
*Grassberger (1999a), based on dw and Eq. (5.13). Series expansions for d = 2 yield § = | / v = 1.32 ± 0.02 (Essam
